- §18.2(i) Definition
- §18.2(ii) $x$-Difference Operators
- §18.2(iii) Standardization and Related Constants
- §18.2(iv) Recurrence Relations
- §18.2(v) Christoffel–Darboux Formula
- §18.2(vi) Zeros
- §18.2(vii) Quadratic Transformations
- §18.2(viii) Uniqueness of Orthogonality Measure and Completeness
- §18.2(ix) Moments
- §18.2(x) Orthogonal Polynomials and Continued Fractions
- §18.2(xi) Some Special Classes of General Orthogonal Polynomials
- §18.2(xii) Other Special Constructions Involving General OP’s

Let $(a,b)$ be a finite or infinite open interval in $\mathbb{R}$. A system (or set)
of polynomials $\{{p}_{n}(x)\}$, $n=0,1,2,\mathrm{\dots}$,
where ${p}_{n}(x)$ has degree $n$ as in §18.1(i),
is said to be
*orthogonal on* $(a,b)$ *with respect to the weight function*
$w(x)$ ($\ge 0$) *if*

18.2.1 | $${\int}_{a}^{b}{p}_{n}(x){p}_{m}(x)w(x)dx=0,$$ | ||

$n\ne m$. | |||

Here $w(x)$ is continuous or piecewise continuous or integrable such that

18.2.1_5 | $$ | ||

$$ | |||

$n\in \mathbb{N}$. | |||

It is assumed throughout this chapter that for each polynomial ${p}_{n}(x)$
that is orthogonal on an open interval $(a,b)$ the variable $x$ is confined to
the closure of $(a,b)$ *unless indicated otherwise.*
(However, under appropriate conditions almost all equations given in the chapter
can be continued analytically to various complex values of the variables.)

Let $X$ be a finite set of distinct points on $\mathbb{R}$, or a countable infinite
set of distinct points on $\mathbb{R}$, and ${w}_{x}$, $x\in X$, be a set of positive
constants. Then a system of polynomials $\{{p}_{n}(x)\}$, $n=0,1,2,\mathrm{\dots}$, is
said to be *orthogonal* on $X$ with respect to the *weights* ${w}_{x}$ if

18.2.2 | $$\sum _{x\in X}{p}_{n}(x){p}_{m}(x){w}_{x}=0,$$ | ||

$n\ne m$, | |||

when $X$ is infinite, or

18.2.3 | $$\sum _{x\in X}{p}_{n}(x){p}_{m}(x){w}_{x}=0,$$ | ||

$n,m=0,1,\mathrm{\dots},N;n\ne m$, | |||

when $X$ is a finite set of $N+1$ distinct points. In the former case we also require

18.2.4 | $$ | ||

$n=0,1,\mathrm{\dots}$, | |||

whereas in the latter case the system $\{{p}_{n}(x)\}$ is finite: $n=0,1,\mathrm{\dots},N$.

More generally than (18.2.1)–(18.2.3), $w(x)dx$ may be replaced in (18.2.1) by $d\mu (x)$, where the measure $\mu $ is the Lebesgue–Stieltjes measure ${\mu}_{\alpha}$ corresponding to a bounded nondecreasing function $\alpha $ on the closure of $(a,b)$ with an infinite number of points of increase, and such that $$ for all $n$. See §1.4(v), McDonald and Weiss (1999, Chapters 3, 4) and Szegő (1975, §1.4). Then

18.2.4_5 | $${\int}_{a}^{b}{p}_{n}(x){p}_{m}(x)d\mu (x)=0,$$ | ||

$n\ne m$. | |||

If the orthogonality discrete set $X$ is $\{0,1,\mathrm{\dots},N\}$ or $\{0,1,2,\mathrm{\dots}\}$, then the role of the differentiation operator $d/dx$ in the case of classical OP’s (§18.3) is played by ${\mathrm{\Delta}}_{x}$, the forward-difference operator, or by ${\nabla}_{x}$, the backward-difference operator; compare §18.1(i). This happens, for example, with the Hahn class OP’s (§18.20(i)).

If the orthogonality interval is $(-\mathrm{\infty},\mathrm{\infty})$ or $(0,\mathrm{\infty})$, then the role of $d/dx$ can be played by ${\delta}_{x}$, the central-difference operator in the imaginary direction (§18.1(i)). This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)).

The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials ${p}_{n}(x)$ uniquely up to constant factors, which may be fixed by suitable standardizations.

Throughout this chapter we will use constants ${h}_{n}$ and ${k}_{n}$, and variants of these, related to OP’s ${p}_{n}(x)$.

The ${h}_{n}$ are defined as:

18.2.5 | ${h}_{n}$ | $={\displaystyle {\int}_{a}^{b}}{\left({p}_{n}(x)\right)}^{2}w(x)dx$ | ||

$\text{or}{h}_{n}$ | $={\displaystyle \sum _{x\in X}}{\left({p}_{n}(x)\right)}^{2}{w}_{x}$ | |||

$\text{or}{h}_{n}$ | $={\displaystyle {\int}_{a}^{b}}{\left({p}_{n}(x)\right)}^{2}d\mu (x).$ | |||

Thus

18.2.5_5 | $${\int}_{a}^{b}{p}_{n}(x){p}_{m}(x)w(x)dx={h}_{n}{\delta}_{n,m},$$ | ||

The constants ${\stackrel{~}{h}}_{n}$, ${k}_{n}$, ${\stackrel{~}{k}}_{n}$ and ${\stackrel{~}{\stackrel{~}{k}}}_{n}$ are defined as:

18.2.6 | ${\stackrel{~}{h}}_{n}$ | $={\displaystyle {\int}_{a}^{b}}x{\left({p}_{n}(x)\right)}^{2}w(x)dx$ | ||

$\text{or}{\stackrel{~}{h}}_{n}$ | $={\displaystyle \sum _{x\in X}}x{\left({p}_{n}(x)\right)}^{2}{w}_{x}$ | |||

$\text{or}{\stackrel{~}{h}}_{n}$ | $={\displaystyle {\int}_{a}^{b}}x{\left({p}_{n}(x)\right)}^{2}d\mu (x),$ | |||

and

18.2.7 | $${p}_{n}(x)={k}_{n}{x}^{n}+{\stackrel{~}{k}}_{n}{x}^{n-1}+{\stackrel{~}{\stackrel{~}{k}}}_{n}{x}^{n-2}+\mathrm{\cdots},$$ | ||

where ${\stackrel{~}{k}}_{0}=0$ and ${\stackrel{~}{\stackrel{~}{k}}}_{n}=0$ for $n=0,1$.

The *classical orthogonal polynomials* are defined with:

(i) the *traditional OP* standardizations of Table 18.3.1, where each is
defined in terms of the above constants.

Two, more specialized, standardizations are:

(ii) *monic OP’s*: ${k}_{n}=1$.

(iii) *orthonormal OP’s*: ${h}_{n}=1$ (and usually, but not always,
${k}_{n}>0$);

As in §18.1(i) we assume that ${p}_{-1}(x)\equiv 0$.

18.2.8 | $${p}_{n+1}(x)=({A}_{n}x+{B}_{n}){p}_{n}(x)-{C}_{n}{p}_{n-1}(x),$$ | ||

$n\ge 0$. | |||

Here ${A}_{n}$, ${B}_{n}$ ($n\ge 0$), and ${C}_{n}$ ($n\ge 1$) are real constants. Then

18.2.9 | ${A}_{n}$ | $={\displaystyle \frac{{k}_{n+1}}{{k}_{n}}},$ | ||

${B}_{n}$ | $=\left({\displaystyle \frac{{\stackrel{~}{k}}_{n+1}}{{k}_{n+1}}}-{\displaystyle \frac{{\stackrel{~}{k}}_{n}}{{k}_{n}}}\right){A}_{n}=-{\displaystyle \frac{{\stackrel{~}{h}}_{n}}{{h}_{n}}}{A}_{n},$ | |||

${C}_{n}$ | $={\displaystyle \frac{{A}_{n}{\stackrel{~}{\stackrel{~}{k}}}_{n}+{B}_{n}{\stackrel{~}{k}}_{n}-{\stackrel{~}{\stackrel{~}{k}}}_{n+1}}{{k}_{n-1}}}={\displaystyle \frac{{A}_{n}}{{A}_{n-1}}}{\displaystyle \frac{{h}_{n}}{{h}_{n-1}}}.$ | |||

Hence ${A}_{n-1}{A}_{n}{C}_{n}={A}_{n}^{2}{h}_{n}/{h}_{n-1}$ ($n\ge 1$), so

18.2.9_5 | $${A}_{n-1}{A}_{n}{C}_{n}>0,$$ | ||

$n\ge 1$. | |||

The OP’s are orthonormal iff ${C}_{n}={A}_{n}/{A}_{n-1}$ ($n\ge 1$) and ${h}_{0}=1$. The OP’s are monic iff ${A}_{n}=1$ ($n\ge 0$) and ${k}_{0}=1$.

18.2.10 | $$x{p}_{n}(x)={a}_{n}{p}_{n+1}(x)+{b}_{n}{p}_{n}(x)+{c}_{n}{p}_{n-1}(x),$$ | ||

$n\ge 0$. | |||

Here ${a}_{n}$, ${b}_{n}$ ($n\ge 0$), ${c}_{n}$ ($n\ge 1$) are real constants. Then

18.2.11 | ${a}_{n}$ | $={\displaystyle \frac{{k}_{n}}{{k}_{n+1}}},$ | ||

${b}_{n}$ | $={\displaystyle \frac{{\stackrel{~}{k}}_{n}}{{k}_{n}}}-{\displaystyle \frac{{\stackrel{~}{k}}_{n+1}}{{k}_{n+1}}}={\displaystyle \frac{{\stackrel{~}{h}}_{n}}{{h}_{n}}},$ | |||

${c}_{n}$ | $={\displaystyle \frac{{\stackrel{~}{\stackrel{~}{k}}}_{n}-{a}_{n}{\stackrel{~}{\stackrel{~}{k}}}_{n+1}-{b}_{n}{\stackrel{~}{k}}_{n}}{{k}_{n-1}}}={a}_{n-1}{\displaystyle \frac{{h}_{n}}{{h}_{n-1}}}.$ | |||

Hence

18.2.11_1 | $$\sum _{j=0}^{n}{b}_{j}=\frac{{\stackrel{~}{k}}_{n+1}}{{k}_{n+1}}.$$ | ||

Furthermore, ${a}_{n-1}{c}_{n}={a}_{n-1}^{2}{h}_{n}/{h}_{n-1}$ ($n\ge 1$), so

18.2.11_2 | $${a}_{n-1}{c}_{n}>0,$$ | ||

$n\ge 1$. | |||

The OP’s are orthonormal iff ${c}_{n}={a}_{n-1}$ ($n\ge 1$) and ${h}_{0}=1$. The OP’s are monic iff ${a}_{n}=1$ ($n\ge 0$) and ${k}_{0}=1$.

The coefficients ${A}_{n},{B}_{n},{C}_{n}$ in the first form and ${a}_{n},{b}_{n},{c}_{n}$ in the second form are related by

18.2.11_3 | ${a}_{n}$ | $={A}_{n}^{-1},{b}_{n}=-{A}_{n}^{-1}{B}_{n},{c}_{n}={A}_{n}^{-1}{C}_{n};$ | ||

${A}_{n}$ | $={a}_{n}^{-1},{B}_{n}=-{a}_{n}^{-1}{b}_{n},{C}_{n}={a}_{n}^{-1}{c}_{n}.$ | |||

Assume that the ${p}_{n}(x)$ are *monic*, so ${A}_{n}=1={a}_{n}$.
Then, with

18.2.11_4 | ${\alpha}_{n}$ | $\equiv {b}_{n}=-{B}_{n},$ | ||

${\beta}_{n}$ | $\equiv {c}_{n}={C}_{n}={h}_{n}/{h}_{n-1},$ | |||

the *monic* recurrence relations (18.2.8)
and (18.2.10) take the form

18.2.11_5 | $x{p}_{n}(x)$ | $={p}_{n+1}(x)+{\alpha}_{n}{p}_{n}(x)+{\beta}_{n}{p}_{n-1}(x),$ | ||

$n\ge 1$, | ||||

${p}_{1}(x)$ | $=x-{\alpha}_{0},$ | |||

${p}_{0}(x)$ | $=1.$ | |||

See also (3.5.30). Note that

18.2.11_6 | $${\beta}_{n}>0,$$ | ||

$n\ge 1$. | |||

In terms of the monic OP’s ${p}_{n}$ define the orthonormal OP’s ${q}_{n}$ by

18.2.11_7 | $${q}_{n}(x)={p}_{n}(x)/\sqrt{{h}_{n}},$$ | ||

$n\ge 0$. | |||

Then, with the coefficients (18.2.11_4) associated
with the monic OP’s ${p}_{n}$,
the *orthonormal* recurrence relation for ${q}_{n}$
takes the form

18.2.11_8 | $x{q}_{n}(x)$ | $=\sqrt{{\beta}_{n+1}}{q}_{n+1}(x)+{\alpha}_{n}{q}_{n}(x)+\sqrt{{\beta}_{n}}{q}_{n-1}(x),$ | ||

$n\ge 1$ | ||||

${q}_{1}(x)$ | $=(x-{\alpha}_{0})/\sqrt{{h}_{0}{\beta}_{1}},$ | |||

${q}_{0}(x)$ | $=1/\sqrt{{h}_{0}},$ | |||

with ${h}_{0}$ still being associated with the monic ${p}_{0}(x)=1$.

If polynomials ${p}_{n}(x)$ are generated by recurrence relation
(18.2.8) under assumption of inequality
(18.2.9_5) (or similarly for the other three forms)
then the ${p}_{n}(x)$ are orthogonal by *Favard’s theorem*, see
§18.2(viii), in that the existence of
a bounded non-decreasing function $\alpha (x)$ on $(a,b)$
yielding the orthogonality realtion (18.2.4_5)
is guaranteed.

If the polynomials ${p}_{n}(x)$ ($n=0,1,\mathrm{\dots},N$) are orthogonal on a finite set $X$ of $N+1$ distinct points as in (18.2.3), then the polynomial ${p}_{N+1}(x)$ of degree $N+1$, up to a constant factor defined by (18.2.8) or (18.2.10), vanishes on $X$.

The recurrence relations (18.2.10) can be equivalently written as

18.2.11_9 | $$\left(\begin{array}{cccc}{b}_{0}& {a}_{0}& & 0\\ {c}_{1}& {b}_{1}& {a}_{1}& \\ & {c}_{2}& \mathrm{\ddots}& \mathrm{\ddots}\\ 0& & \mathrm{\ddots}& \mathrm{\ddots}\end{array}\right)\left(\begin{array}{c}{p}_{0}(x)\\ {p}_{1}(x)\\ \mathrm{\vdots}\\ \mathrm{\vdots}\end{array}\right)=x\left(\begin{array}{c}{p}_{0}(x)\\ {p}_{1}(x)\\ \mathrm{\vdots}\\ \mathrm{\vdots}\end{array}\right).$$ | ||

The matrix on the left-hand side is an (infinite tridiagonal)
*Jacobi matrix*. This matrix is symmetric iff ${c}_{n}={a}_{n-1}$
($n\ge 1$).

With notation (18.2.4_5), (18.2.5), (18.2.7)

18.2.12 | $${K}_{n}(x,y)\equiv \sum _{\mathrm{\ell}=0}^{n}\frac{{p}_{\mathrm{\ell}}(x){p}_{\mathrm{\ell}}(y)}{{h}_{\mathrm{\ell}}}=\frac{{k}_{n}}{{h}_{n}{k}_{n+1}}\frac{{p}_{n+1}(x){p}_{n}(y)-{p}_{n}(x){p}_{n+1}(y)}{x-y},$$ | ||

$x\ne y$, | |||

18.2.12_5 | $${\int}_{a}^{b}f(y){K}_{n}(x,y)d\mu (y)=\{\begin{array}{cc}f(x),\hfill & f\in \mathrm{Span}({p}_{0},{p}_{1},\mathrm{\dots},{p}_{n})\text{,}\hfill \\ 0,\hfill & {\int}_{a}^{b}f(x){p}_{\mathrm{\ell}}(x)d\mu (x)=0\text{(}\mathrm{\ell}=0,1,\mathrm{\dots},n\text{).}\hfill \end{array}$$ | ||

18.2.13 | $${K}_{n}(x,x)=\sum _{\mathrm{\ell}=0}^{n}\frac{{({p}_{\mathrm{\ell}}(x))}^{2}}{{h}_{\mathrm{\ell}}}=\frac{{k}_{n}}{{h}_{n}{k}_{n+1}}\left({p}_{n+1}^{\prime}(x){p}_{n}(x)-{p}_{n}^{\prime}(x){p}_{n+1}(x)\right).$$ | ||

Assume $y\notin (a,b)$ in (18.2.12).
Then the *kernel polynomials*

18.2.14 | $${q}_{n}(x)={q}_{n}(x;y)\equiv {K}_{n}(x,y)=\sum _{\mathrm{\ell}=0}^{n}\frac{{p}_{\mathrm{\ell}}(x){p}_{\mathrm{\ell}}(y)}{{h}_{\mathrm{\ell}}}$$ | ||

are OP’s with orthogonality relation

18.2.15 | $${\int}_{a}^{b}{q}_{n}(x){q}_{m}(x)|y-x|d\mu (x)=0,$$ | ||

$n\ne m$. | |||

Between the systems $\{{p}_{n}(x)\}$ and $\{{q}_{n}(x)\}$ there are the contiguous relations

18.2.16 | ${q}_{n}(x)-{q}_{n-1}(x)$ | $={\displaystyle \frac{{p}_{n}(y)}{{h}_{n}}}{p}_{n}(x),$ | ||

18.2.17 | ${p}_{n}(y){p}_{n+1}(x)-{p}_{n+1}(y){p}_{n}(x)$ | $={\displaystyle \frac{{h}_{n}{k}_{n+1}}{{k}_{n}}}(x-y){q}_{n}(x).$ | ||

All $n$ zeros of an OP ${p}_{n}(x)$ are simple, and they are located in the interval of orthogonality $(a,b)$. The zeros of ${p}_{n}(x)$ and ${p}_{n+1}(x)$ separate each other, and if $$ then between any two zeros of ${p}_{m}(x)$ there is at least one zero of ${p}_{n}(x)$.

For usage of the zeros of an OP in Gauss quadrature see §3.5(v). When the Jacobi matrix in (18.2.11_9) is truncated to an $n\times n$ matrix

18.2.18 | $${\mathbf{J}}_{n}=\left(\begin{array}{ccccc}{b}_{0}& {a}_{0}& & & 0\\ {c}_{1}& {b}_{1}& {a}_{1}& & \\ & {c}_{2}& \mathrm{\ddots}& \mathrm{\ddots}& \\ & & \mathrm{\ddots}& \mathrm{\ddots}& {a}_{n-2}\\ 0& & & {c}_{n-1}& {b}_{n-1}\end{array}\right)$$ | ||

then the zeros of ${p}_{n}(x)$ are the eigenvalues of ${\mathbf{J}}_{n}$ (see also §3.5(vi)).

Let ${x}_{1},\mathrm{\dots},{x}_{n}$ be the zeros of the OP ${p}_{n}$, so

18.2.19 | $${p}_{n}(x)={k}_{n}\prod _{j=1}^{n}(x-{x}_{j}).$$ | ||

The *discriminant* of ${p}_{n}$ is defined by

18.2.20 | $$ | ||

See Ismail (2009, §3.4) for another expression of the discriminant in the case of a general OP.

For OP’s $\{{p}_{n}(x)\}$ on $\mathbb{R}$ with respect to an *even*
weight function $w(x)$ we have

18.2.21 | $${p}_{n}(-x)={(-1)}^{n}{p}_{n}(x),$$ | ||

so we can put

18.2.22 | ${r}_{n}({x}^{2})$ | $\equiv {p}_{2n}(x),$ | ||

${s}_{n}({x}^{2})$ | $\equiv {x}^{-1}{p}_{2n+1}(x),$ | |||

$v({x}^{2})$ | $\equiv w(x).$ | |||

Then $\{{r}_{n}(x)\}$ are OP’s on $(0,\mathrm{\infty})$ with respect to weight function ${x}^{-\frac{1}{2}}v(x)$ and $\{{s}_{n}(x)\}$ are OP’s on $(0,\mathrm{\infty})$ with respect to weight function ${x}^{\frac{1}{2}}v(x)$.

As a slight variant let $\{{p}_{n}(x)\}$ be OP’s with respect to an
*even* weight function $w(x)$ on $(-1,1)$.
Then (18.2.21) still holds and we can put

18.2.23 | ${r}_{n}(2{x}^{2}-1)$ | $\equiv {p}_{2n}(x),$ | ||

${s}_{n}(2{x}^{2}-1)$ | $\equiv {x}^{-1}{p}_{2n+1}(x),$ | |||

$v(2{x}^{2}-1)$ | $\equiv w(x).$ | |||

Then $\{{r}_{n}(x)\}$ are OP’s on $(-1,1)$ with respect to weight function ${(1+x)}^{-\frac{1}{2}}v(x)$ and $\{{s}_{n}(x)\}$ are OP’s on $(-1,1)$ with respect to weight function ${(1+x)}^{\frac{1}{2}}v(x)$.

See Chihara (1978, Ch. I, §8).

If a system of polynomials $\{{p}_{n}(x)\}$ satisfies any of the formula pairs
(recurrence relation and coefficient inequality)
(18.2.8), (18.2.9_5) or
(18.2.10), (18.2.11_2) or
(18.2.11_5), (18.2.11_6) or
(18.2.11_8), (18.2.11_6)
then $\{{p}_{n}(x)\}$
is orthogonal with respect to some positive measure on $\mathbb{R}$
(*Favard’s theorem*). The measure is not necessarily
absolutely continuous (i.e., of the form $w(x)dx$)
nor is it necessarily unique,
up to a positive constant factor. However,
if OP’s have an orthogonality relation on a bounded interval,
then their orthogonality measure is unique, up to a positive constant factor.

A system $\{{p}_{n}(x)\}$ of OP’s satisfying (18.2.1) and (18.2.5) is complete if each $f(x)$ in the Hilbert space ${L}_{w}^{2}((a,b))$ can be approximated in Hilbert norm by finite sums ${\sum}_{n}{\lambda}_{n}{p}_{n}(x)$. For such a system, functions $f\in {L}_{w}^{2}((a,b))$ and sequences $\{{\lambda}_{n}\}$ ($n=0,1,2,\mathrm{\dots}$) satisfying $$ can be related to each other in a similar way as was done for Fourier series in (1.8.1) and (1.8.2):

18.2.24 | $${\lambda}_{n}={h}_{n}^{-1}{\int}_{a}^{b}f(x){p}_{n}(x)w(x)dx$$ | ||

if and only if

18.2.25 | $$f(x)=\sum _{n=0}^{\mathrm{\infty}}{\lambda}_{n}{p}_{n}(x)$$ | ||

(convergence in ${L}_{w}^{2}((a,b))$). A system of OP’s with unique orthogonality measure is always complete, see Shohat and Tamarkin (1970, Theorem 2.14). In particular, a system of OP’s on a bounded interval is always complete.

The moments for an orthogonality measure $d\mu (x)$ are the numbers

18.2.26 | $${\mu}_{n}={\int}_{a}^{b}{x}^{n}d\mu (x),$$ | ||

$n=0,1,2,\mathrm{\dots}$. | |||

The Hankel determinant ${\mathrm{\Delta}}_{n}$ of order $n$ is defined by ${\mathrm{\Delta}}_{0}=1$ and

18.2.27 | $${\mathrm{\Delta}}_{n}=\left|\begin{array}{cccc}{\mu}_{0}& {\mu}_{1}& \mathrm{\dots}& {\mu}_{n-1}\\ {\mu}_{1}& {\mu}_{2}& \mathrm{\dots}& {\mu}_{n}\\ \mathrm{\vdots}& \mathrm{\vdots}& & \mathrm{\vdots}\\ {\mu}_{n-1}& {\mu}_{n}& \mathrm{\dots}& {\mu}_{2n-2}\end{array}\right|,$$ | ||

$n=1,2,\mathrm{\dots}$. | |||

Also define determinants ${\mathrm{\Delta}}_{n}^{\prime}$ by ${\mathrm{\Delta}}_{0}^{\prime}=0$, ${\mathrm{\Delta}}_{1}^{\prime}={\mu}_{1}$ and

18.2.28 | $${\mathrm{\Delta}}_{n}^{\prime}=\left|\begin{array}{ccccc}{\mu}_{0}& {\mu}_{1}& \mathrm{\dots}& {\mu}_{n-2}& {\mu}_{n}\\ {\mu}_{1}& {\mu}_{2}& \mathrm{\dots}& {\mu}_{n-1}& {\mu}_{n+1}\\ \mathrm{\vdots}& \mathrm{\vdots}& & \mathrm{\vdots}& \mathrm{\vdots}\\ {\mu}_{n-1}& {\mu}_{n}& \mathrm{\dots}& {\mu}_{2n-3}& {\mu}_{2n-1}\end{array}\right|,$$ | ||

$n=2,3,\mathrm{\dots}$. | |||

The monic OP’s ${p}_{n}(x)$ with respect to the measure $d\mu (x)$ can be expressed in terms of the moments by

18.2.29 | $${p}_{n}(x)=\frac{1}{{\mathrm{\Delta}}_{n}}\left|\begin{array}{cccc}{\mu}_{0}& {\mu}_{1}& \mathrm{\dots}& {\mu}_{n}\\ {\mu}_{1}& {\mu}_{2}& \mathrm{\dots}& {\mu}_{n+1}\\ \mathrm{\vdots}& \mathrm{\vdots}& & \mathrm{\vdots}\\ {\mu}_{n-1}& {\mu}_{n}& \mathrm{\dots}& {\mu}_{2n-1}\\ 1& x& \mathrm{\dots}& {x}^{n}\end{array}\right|,$$ | ||

$n=1,2,\mathrm{\dots}$. | |||

The recurrence coefficients ${\alpha}_{n}$ and ${\beta}_{n}$ in (18.2.11_5) can be expressed in terms of the determinants (18.2.27) and (18.2.28) by

18.2.30 | ${\alpha}_{n}$ | $={\displaystyle \frac{{\mathrm{\Delta}}_{n+1}^{\prime}}{{\mathrm{\Delta}}_{n+1}}}-{\displaystyle \frac{{\mathrm{\Delta}}_{n}^{\prime}}{{\mathrm{\Delta}}_{n}}},$ | ||

$n=0,1,2,\mathrm{\dots}$. | ||||

${\beta}_{n}$ | $={\displaystyle \frac{{\mathrm{\Delta}}_{n+1}{\mathrm{\Delta}}_{n-1}}{{\mathrm{\Delta}}_{n}^{2}}},$ | |||

$n=1,2,\mathrm{\dots}$. | ||||

It is to be noted that, although formally correct, the results of (18.2.30) are of little utility for numerical work, as Hankel determinants are notoriously ill-conditioned. See Gautschi (2004, p. 54), and Golub and Meurant (2010, pp. 56, 57). Alternatives for numerical calculation of the recursion coefficients in terms of the moments are discussed in these references, and in §18.40(ii).

In this subsection fix the recurrence coefficients ${\alpha}_{n}$ ($n=0,1,2,\mathrm{\dots}$) and ${\beta}_{n}$ ($n=1,2,\mathrm{\dots}$) as in (18.2.11_5), with ${p}_{n}(x)$ the corresponding monic OP’s and with $d\mu (x)$, $a$ and $b$ as in the orthogonality relation (18.2.4_5). Define the first associated monic orthogonal polynomials ${p}_{n}^{(1)}(x)$ as monic OP’s satisfying

18.2.31 | ${p}_{0}^{(1)}(x)$ | $=1,$ | ||

${p}_{1}^{(1)}(x)$ | $=x-{\alpha}_{1},$ | |||

$x{p}_{n}^{(1)}(x)$ | $={p}_{n+1}^{(1)}(x)+{\alpha}_{n+1}{p}_{n}^{(1)}(x)+{\beta}_{n+1}{p}_{n-1}^{(1)}(x),$ | |||

$n=1,2,\mathrm{\dots}$, | ||||

where the *first* indicates that the indices of the recursion
coefficients ${\alpha}_{n}$, ${\beta}_{n}$ of
(18.2.31)
have been incremented by $1$, when compared to those of
(18.2.11_5).
More generally, §18.30 defines the recurrence
relation of the $c$th associated monic OP
by means of a similar shift by $c$ in (18.2.11_5).

The OP’s ${p}_{n}^{(1)}(x)$ may also be calculated from the original recursion (18.2.11_5), but with independent initial conditions for ${p}_{0},{p}_{1}$:

18.2.32 | ${p}_{0}^{(0)}(x)$ | $=0,$ | ||

${p}_{1}^{(0)}(x)$ | $=1,$ | |||

$x{p}_{n}^{(0)}(x)$ | $={p}_{n+1}^{(0)}(x)+{\alpha}_{n}{p}_{n}^{(0)}(x)+{\beta}_{n}{p}_{n-1}^{(0)}(x),$ | |||

$n=1,2,\mathrm{\dots}$, | ||||

resulting in ${p}_{n}^{(0)}(x)={p}_{n-1}^{(1)}(x)$, by simple comparison of
the two recursions. The ${p}_{n}^{(0)}(x)$ are the
*monic corecursive orthogonal polynomials*.
These relationships are further explored in
§§18.30(vi) and 18.30(vii).

The polynomials ${p}_{n}^{(1)}(x)$ may be also be directly expressed in terms of the ${p}_{n}(x)$ of (18.2.11_5):

18.2.33 | $${p}_{n-1}^{(1)}(z)=\frac{1}{{\mu}_{0}}{\int}_{a}^{b}\frac{{p}_{n}(z)-{p}_{n}(x)}{z-x}d\mu (x),$$ | ||

$z\in \u2102\backslash [a,b]$, $n=1,2,\mathrm{\dots}$, | |||

with moment ${\mu}_{0}$ defined in (18.2.26).

Using the terminology of §1.12(ii), the $n$-th approximant of the continued fraction

18.2.34 | $$\frac{1}{x-{\alpha}_{0}-}\frac{{\beta}_{1}}{x-{\alpha}_{1}-}\frac{{\beta}_{2}}{x-{\alpha}_{2}-}\mathrm{\cdots}$$ | ||

is given by

18.2.35 | $${F}_{n}(x)=\frac{1}{x-{\alpha}_{0}-}\frac{{\beta}_{1}}{x-{\alpha}_{1}-}\frac{{\beta}_{2}}{x-{\alpha}_{2}-}\mathrm{\cdots}\frac{{\beta}_{n-1}}{x-{\alpha}_{n-1}}.$$ | ||

Then

18.2.36 | $${F}_{n}(x)=\frac{{p}_{n-1}^{(1)}(x)}{{p}_{n}(x)}=\frac{{p}_{n}^{(0)}(x)}{{p}_{n}(x)}=\frac{1}{{\mu}_{0}}\sum _{k=1}^{n}\frac{{w}_{k}}{x-{x}_{k}},$$ | ||

where ${x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}$ are the zeros of ${p}_{n}(x)$ and

18.2.37 | $${w}_{k}={\int}_{a}^{b}\frac{{p}_{n}(x)}{(x-{x}_{k}){p}_{n}^{\prime}({x}_{k})}d\mu (x),$$ | ||

$k=1,2,\mathrm{\dots},n$, | |||

are the Christoffel numbers, see also (3.5.18). Because of (18.2.36) the OP’s ${p}_{n}(x)$ are also called monic denominator polynomials and the OP’s ${p}_{n-1}^{(1)}(x)$, or, equivalently, the ${p}_{n}^{(0)}(x)$, are called the monic numerator polynomials.

Assume that the interval $[a,b]$ is bounded. Markov’s theorem states that

18.2.38 | $$\underset{n\to \mathrm{\infty}}{lim}{F}_{n}(z)=\frac{1}{{\mu}_{0}}{\int}_{a}^{b}\frac{d\mu (x)}{z-x},$$ | ||

$z\in \u2102\backslash [a,b]$. | |||

See Chihara (1978, pp. 86–89), and, in slightly different notation,
Ismail (2009, §§2.3, 2.6, 2.10), where it is
assumed that ${\mu}_{0}=1$.
See also the extended development of these ideas in
§§18.30(vi), 18.30(vii), and in §18.40(ii) where they form the basis for one method of solving the *classical moment problem*.

This is the class of weight functions $w$ on $(-1,1)$ such that, in addition to (18.2.1_5),

18.2.39 | $$ | ||

For OP’s ${p}_{n}(x)$ with weight function in the class $\mathcal{G}$
there are
asymptotic formulas as $n\to \mathrm{\infty}$, respectively for $x\in \u2102$
outside $[-1,1]$ and for $x\in [-1,1]$, see
Szegő (1975, Theorems 12.1.2, 12.1.4).
Under further conditions on the weight function there is an
*equiconvergence theorem*, see
Szegő (1975, Theorem 13.1.2).
This says roughly that
the series (18.2.25) has the same pointwise
convergence behavior as the same series with
${p}_{n}(x)={T}_{n}\left(x\right)$, a Chebyshev polynomial of the first
kind, see Table 18.3.1.

Nevai (1979, p.39) defined the class $\mathcal{S}$ of orthogonality measures with support inside $[-1,1]$ such that the absolutely continuous part $w(x)dx$ has $w$ in the Szegő class $\mathcal{G}$. For OP’s with orthogonality measure in $\mathcal{S}$ Nevai (1979, pp. 148–150) generalized Szegő’s equiconvergence theorem. In further generalizations of the class $\mathcal{S}$ discrete mass points ${x}_{k}$ outside $[-1,1]$ are allowed. If these ${x}_{k}$ satisfy $$ then Szegő type asymptotics outside $[-1,1]$ can be given for the corresponding OP’s, see Simon (2011, Corollary 3.7.2 and following).

The class $\mathbf{M}(a,b)$ ($a>0$, $b\in \mathbb{R}$), introduced by Nevai (1979, p.10), consists of all orthogonality measures $d\mu $ such that the coefficients $\sqrt{{\beta}_{n}}$ and ${\alpha}_{n}$ in the recurrence relation (18.2.11_8) for the corresponding orthonormal OP’s satisfy

18.2.40 | $\underset{n\to \mathrm{\infty}}{lim}\sqrt{{\beta}_{n}}$ | $=\frac{1}{2}a,$ | ||

$\underset{n\to \mathrm{\infty}}{lim}{\alpha}_{n}$ | $=b.$ | |||

If $d\mu \in \mathbf{M}(a,b)$ then the interval $[b-a,b+a]$ is included in the support of $d\mu $, and outside $[b-a,b+a]$ the measure $d\mu $ only has discrete mass points ${x}_{k}$ such that $b\pm a$ are the only possible limit points of the sequence $\{{x}_{k}\}$, see Máté et al. (1991, Theorem 10). Part of this theorem was already proved by Blumenthal (1898). Therefore this class is also called the Nevai–Blumenthal class.

For OP’s ${p}_{n}$ with ${h}_{n}$ and orthogonality relation
as in (18.2.5)
and (18.2.5_5), the
*Poisson kernel* is defined by

18.2.41 | $${P}_{z}(x,y)=\sum _{n=0}^{\mathrm{\infty}}\frac{{p}_{n}(x){p}_{n}(y)}{{h}_{n}}{z}^{n},$$ | ||

$$, | |||

for $x,y$ in the support of the orthogonality measure and $z$ such that the series in (18.2.41) converges absolutely for all these $x,y$. Instances where the Poisson kernel is nonnegative are of special interest, see Ismail (2009, Theorem 4.7.12).

For a large class of OP’s ${p}_{n}$ there exist pairs of differentiation formulas

18.2.42 | ${\pi}_{n}(x){p}_{n}^{\prime}(x)+{A}_{n}(x){p}_{n}(x)$ | $={\lambda}_{n}{p}_{n-1}(x),$ | ||

18.2.43 | ${\pi}_{n}(x){p}_{n}^{\prime}(x)+{B}_{n}(x){p}_{n}(x)$ | $={\mu}_{n}{p}_{n+1}(x),$ | ||

see Ismail (2009, (3.2.3), (3.2.10)). If ${A}_{n}(x)$ and ${B}_{n}(x)$ are polynomials of degree independent of $n$, and moreover ${\pi}_{n}(x)$ is a polynomial $\pi (x)$ independent of $n$ then

18.2.44 | $$\pi (x){p}_{n}^{\prime}(x)=\sum _{j=n-s}^{n+t}{a}_{n,j}{p}_{j}(x)$$ | ||

for certain coefficients ${a}_{n,j}$ with $s,t$ independent of $n$.
Then the OP’s are called *semi-classical* and
(18.2.44) is called a *structure relation*.

Polynomials ${p}_{n}(x)$ of degree $n$ ($n=0,1,2,\mathrm{\dots}$) are called
*Sheffer polynomials* if they are generated by a generating function
of the form

18.2.45 | $$f(t)\mathrm{exp}\left(xu(t)\right)=\sum _{n=0}^{\mathrm{\infty}}{c}_{n}{p}_{n}(x)\frac{{t}^{n}}{n!},$$ | ||

where $f(t)$ and $u(t)$ are formal power series in $t$, with $f(0)=1$, $u(0)=0$ and ${u}^{\prime}(0)=1$. Often a standardization ${c}_{n}=1$ is taken. If $v(s)$ is the formal power series such that $v(u(t))=t$ then a property equivalent to (18.2.45) with ${c}_{n}=1$ is that

18.2.46 | $$v\left({D}_{x}\right){p}_{n}(x)=n{p}_{n-1}(x).$$ | ||

The operator ${D}_{x}$ is a *delta operator*, i.e., ${D}_{x}$ commutes with translation in the variable $x$ and ${D}_{x}x$ is a nonzero constant.

The generating functions (18.12.13), (18.12.15), (18.23.3), (18.23.4), (18.23.5) and (18.23.7) for Laguerre, Hermite, Krawtchouk, Meixner, Charlier and Meixner–Pollaczek polynomials, respectively, can be written in the form (18.2.45). In fact, these are the only OP’s which are Sheffer polynomials (with Krawtchouk polynomials being only a finite system)